nLab
inductive-recursive type

Contents

Context

Deduction and Induction

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Induction

Contents

Idea

In type theory, induction-recursion is a principle for mutually defining types of the form

A:Type,andB:AType, A \colon Type\,,\;\;\; and \;\;\; B \colon A \to Type \,,

where AA is defined as an inductive type and BB is defined by recursion on AA. Crucially, the definition of AA may use BB. Without this last requirement, we could first define AA and then separately BB.

Example

(from ForsSetz)

The universe a la Tarski is an example of an inductive-recursive definition, where a set UU is defined inductively together with a recursive function T:USetT: U \to Set. The constructors for UU may depend negatively on TT applied to elements of UU, as is the case if UU, for example, is closed under dependent function spaces:

a:Ub:T(a)Uπ(a,b):U \frac{a:U \;\;\;\;\;\;b : T(a) \to U}{\pi(a, b) : U}

with T(π(a,b))= x:T(a)T(b(x))T(\pi(a, b)) = \prod_{x:T(a)} T(b(x)).

Here, T:USetT:U \to Set is defined recursively. Sometimes, however, one might not want to give T(u)T(u) completely as soon as u:Uu:U is introduced, but instead define TT inductively as well. This is the principle of induction-induction.

References

Last revised on October 24, 2016 at 06:24:12. See the history of this page for a list of all contributions to it.